1. PDF from Hadronic Tensor on the Lattice and Connected Sea Evolution
Path-integral Formulation of Hadronic Tensor in DIS
Parton Degrees of Freedom
Evolution of Connected Sea Partons
Numerical Challenges
Workshop on PDF at Beida, July 14-16, 2017

2. Experimental Data
New Muon Collaboration (NMC – PRL 66, 2712 (1991)) μ+ p(n)  μX
Quark parton model + Isospin symmetry
NMC :
asymmetry from Drell-Yan Production (PRL 69, 1726 (1992))
NuTeV experiment (PRL 88, (2002))

3. Hadronic Tensor in Euclidean Path-Integral Formalism
Deep inelastic scattering
In Minkowski space
Euclidean path-integral
KFL and S.J. Dong, PRL 72, 1790 (1994)
KFL, PRD 62, (2000)

4. Wμν in Euclidean Space
Laplace transform

5. Cat’s ears diagrams are suppressed by O(1/Q2).

6. Large momentum frame u d s
Parton degrees of freedom: valence, connected sea and disconnected sea
u d s

7. Properties of this separation
No renormalization
Gauge invariant
Topologically distinct as far as the quark lines are concerned
W1(x, Q2) and W2(x, Q2) are frame independent.
Small x behavior of CS and DS are different.
Short distance expansion (Taylor expansion) OPE

8. Operator Product Expansion -> Taylor Expansion
Dispersion relation
Expand in the unphysical region

9. Euclidean path-integral
Consider
Short-distance expansion ( )
Laplace transform

10. Dispersion relation
Expansion about the unphysical region ( )
even + odd n terms

12. Gottfried Sum Rule Violation
NMC:
two flavor traces ( ) one flavor trace ( )
K.F. Liu and S.J. Dong, PRL 72, 1790 (1994)

13. Comments
The results are the same as derived from the conventional operator product expansion.
Contrary to conventional OPE, the path-integral formalism admits separation of CI and DI.
The OPE turns out to be Taylor expansion of functions in the path-integral formalism.
For with definite n, there is only one CI and one DI in the three-point function, i.e. (a’) is the same as (b’). Thus, one cannot separate quark contribution from that of antiquark in matrix elements.

14. 3) Fitting of experimental data
K.F. Liu, PRD 62, (2000)
3) Fitting of experimental data
But
A better fit
where
4) Unlike DS, CS evolves the same way as the valence.

15. Connected Sea Partons expt CT10 lattice
K.F. Liu, W.C. Chang, H.Y. Cheng, J.C. Peng, PRL 109, (2012)
expt
CT10
lattice

17. Lattice input to global fitting of PDF
<x>u/d (DI)
<x>s
Lattice calculation with Overlap fermion on 3 lattices including on at mπ ~ 140 MeV (Mingyang Sun, )
<x>s= 0.050(16), <x>u/d (DI=0.060(17) <x>s /<x>u/d(DI) = 0.83(7)

18. <x>s /<x>u/d(DI) 0.83(7)
Lattice input to global fitting of PDF
<x>s /<x>u/d(DI) 0.83(7)

19. Operator Mixing
Connected insertion
Disconnected insertion

20. Evolution Equations NNLO Valence u can evolve into valence d ? Note:
S. Moch et al., hep/ ,
Cafarella et al.,
NNLO
Valence u can evolve into valence d ?
Note:

21. Evolution equations separating CS from the DS partons:
11 equations for the general case
K.F. Liu

22. Hadronic Tensor on the Lattice (Inverse Laplace Transform)
Improved Maximum Entropy method
Backus-Gilbert method
Fitting with a prescribed functional form for the spectral distribution.
OPE w/o OPE

23. Improved Maximum Entropy Method
Inverse problem
Bayes’ theorem
Maximum entropy method: find from
Improved MEM (Burnier and Rothkpf, PRL 111, (2013))

24. Frank X. Lee

25. Photo-proton Inclusive X-section

26. Numerical Challenges Bjorken x Range of x:
Lattice calculation of the hadronic tensor – no renormalization, continuum and chiral limits, direct comparison with expts PDF.
Bjorken x
Range of x:

27. Backus-Gilbert reconstruction test
Model data with two peaks at 0.1 and 0.3,
and a continuum with height = 50

28. 123 x 128 lattice (CLQCD), a ~ 0.15 fm, mπ ~ 600 MeV

32. Quark Parton Model

33. OPE w/o OPE U. Aglietti, et al., PLB 432, 411 (1998)
A. J. Chambers et al. (QCDSF), PRL 118, (2017)
Can show that

34. ?
Limited to LO

35. Conclusion
The connected sea partons (CSP) found in path-integral formulation are extracted by combining PDF, experimental data and ratio of lattice matrix elements.
It would be better to have separate evolution equations for the CSP and DSP. The separation will remain at different
This way one can facilitate the comparison with lattice calculation of moments in the CI and DI to the corresponding moments from PDF.
Lattice calculation of hadronic tensor is numerically tougher, but theoretical interpretation is relatively easy. No renormalization is needed and the structure function is frame independent.
Hadronic tensor on the lattice is being tackled.

37. Comments CS and DS are explicitly separated, leading to more equations
(11 vs 7) which can accommodate
There is no flavor-changing evolution of the valence partons.
is the sum of two equations
Once the CS is separated at one Q2 , it will remain separated
at other Q2.
Gluons can split into DS, but not to valence and CS.
It is necessary to separate out CS from DS when quark and antiquark annihilation (higher twist) is included in the evolution eqs (Annihilation involves only DS.)